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a) the thing a person keeps their cash in
b) the thing a person keeps coins in
c) the thing a person keeps the aforementioned things in

and do these terms vary depending on the gender of the person in question?

Myself, I call them
a) a wallet (or possibly a purse, if it is basically a little bag with a fastening, as in picture above);
b) a purse (also as in picture above);
c) a bag, bilum or handbag.

I tend to avoid the words purse and handbag when referring to men’s accoutrements, since some consider those words apply only to women’s accoutrements and become uncomfortable and/or offended. And if I’m going to make people uncomfortable and/or offended, I’d rather it was intentional.